Time consistent portfolio strategies for a general utility function

1. Problem Statement

The paper addresses the Merton portfolio optimization problem in a complete financial market where an investor seeks to maximize expected utility from consumption and terminal wealth. The core challenge addressed is time inconsistency arising from a non-constant time discount rate.

• Market Model: A standard Black-Scholes setting with a risk-free asset (constant rate $r$) and a risky asset following a geometric Brownian motion with constant coefficients ($\mu, \sigma$).
• The Inconsistency: When the discount function $h(t, s)$ is not exponential (i.e., the discount rate $\rho$ varies with time), the standard dynamic programming principle fails. An optimal strategy derived at time $t=0$ is no longer optimal at a future time $t > 0$ because the agent's preferences change.
• Objective: To find subgame perfect strategies (time-consistent strategies) where the agent, at any time $t$, chooses a strategy that remains optimal for their future selves, assuming those future selves will also act optimally.

2. Methodology

The author employs a combination of Stochastic Control Theory, Partial Differential Equations (PDEs), and Fixed Point Theory.

A. Extended Hamilton-Jacobi-Bellman (HJB) Equation

Following the framework of Björk, Murgoci, and Zhou, the paper utilizes an extended HJB system. Unlike the standard HJB, the extended version includes a non-local term representing the time-inconsistency.

• The value function $V(t, x)$ is defined such that it satisfies the extended HJB equation along with a verification theorem.
• The optimal controls (consumption $c$ and portfolio weight $\pi$) are derived in feedback form using the first-order conditions of the extended HJB.

B. Marginal Utility Approach

Instead of solving for the value function $V(t, x)$ directly, the paper works with the marginal value function $v(t, x) = \frac{\partial V}{\partial x}(t, x)$.

• This transformation is crucial because it allows the authors to handle general utility functions $U$ (satisfying Inada conditions and bounded relative risk aversion) more effectively.
• The authors introduce a change of variables using the inverse marginal utility function $I = (U')^{-1}$ and a logarithmic transformation of the state space to linearize certain components of the PDE.

C. The Utility-Weighted Discount Rate

A central theoretical innovation is the definition of a utility-weighted discount rate, denoted as $\rho(t, x)$.

• This rate is not exogenous but is an endogenous variable that depends on the current wealth level and time.
• It is defined as a weighted average of the original discount rate, where the weights are the marginal utilities of future consumption and terminal wealth.
• Mathematically, $\rho(t, x)$ acts as a "correction" term that restores the time-consistency of the problem.

D. Fixed Point Iteration

The paper demonstrates that finding the time-consistent strategy reduces to solving a fixed point problem for the function $\rho(t, x)$ (or its transformed version $\bar{\rho}(t, y)$).

• An operator $F$ is defined such that a fixed point $\bar{\rho} = F[\bar{\rho}]$ yields the correct discount rate structure.
• The existence and uniqueness of this fixed point are proven using contraction mapping arguments on a specific space of bounded functions with bounded derivatives, utilizing estimates from parabolic PDE theory (Feynman-Kac formulas and fundamental solution bounds).

3. Key Contributions

1. General Utility Framework: Unlike previous works restricted to specific utility forms (like CRRA), this paper provides a solution for a general class of utility functions satisfying standard regularity and risk aversion conditions.
2. Marginal Utility Weighted Discount Rate: The paper introduces a novel economic interpretation: the time-inconsistency problem can be solved by replacing the exogenous discount rate with an endogenous utility-weighted discount rate. This rate effectively "internalizes" the future preference changes.
3. Computational Algorithm: The paper provides a constructive algorithm (Step 1 to Step 3) to compute the strategies:
   • Find the fixed point $\bar{\rho}$ via successive approximations.
   • Solve a linear parabolic PDE (via Feynman-Kac) to find the inverse marginal value function $\bar{p}$.
   • Recover the optimal consumption and investment ratios in feedback form.
4. Efficiency: The authors note that the fixed point $\rho$ can be computed efficiently via Monte-Carlo simulation, offering a practical advantage over solving high-dimensional non-linear PDEs directly.

4. Main Results

• Theorem 2.2 (Extended HJB): Establishes that if a $C^{1,2}$ solution to the extended HJB exists, the subgame perfect strategies are given by:
  $$\bar{c}(t, x) = \frac{I(V_x(t, x))}{x}, \quad \bar{\pi}(t, x) = -\frac{\theta V_x(t, x)}{\sigma x V_{xx}(t, x)}$$
• Theorem 2.13 (Fixed Point Existence): Proves that the operator $F$ mapping a candidate discount rate function to a new one is a contraction on a specific function space, guaranteeing a unique fixed point $\bar{\rho}$.
• Theorem 2.17 (Value Function Construction): Shows that the function $G(t, x)$, constructed using the fixed point $\bar{\rho}$ and the associated wealth process, is indeed the value function satisfying the extended HJB.
• Feedback Formulas: The optimal strategies are explicitly given as:
  $$\bar{\pi}(t, x) = -\frac{\theta \bar{p}_y(t, \log v(t, x))}{\sigma \bar{p}(t, \log v(t, x))}$$
  $$\bar{c}(t, x) = \frac{I(v(t, x))}{x}$$
  where $\bar{p}$ is the solution to a linear PDE driven by the fixed point discount rate.

5. Significance

• Theoretical Advancement: The paper bridges the gap between time-inconsistent control theory and general utility maximization. It moves beyond specific utility cases (like power utility) to a general framework, proving that time-consistent strategies exist and are unique under broad conditions.
• Economic Intuition: By defining the "utility-weighted discount rate," the paper offers a clear economic mechanism for how an agent should adjust their time preferences to remain consistent. It suggests that the effective discount rate is a dynamic function of the agent's wealth and marginal utility.
• Practical Application: The proposed algorithm allows for the numerical implementation of these strategies. Since the core difficulty (the fixed point) can be solved via Monte-Carlo methods, the approach is scalable and applicable to real-world portfolio management problems where discount rates are not constant (e.g., hyperbolic discounting or regime-switching preferences).

In summary, Mbodji provides a rigorous mathematical framework to resolve time inconsistency in portfolio management by transforming the problem into a fixed point iteration for a marginal-utility-weighted discount rate, applicable to a wide range of investor preferences.